# Publications of Steve Butler

### Preprints

Ordered multiplicity inverse eigenvalue problem for graphs on six vertices, (with John Ahn, Christine Alar, Beth Bjorkman, Joshua Carlson, Audrey Goodnight, Haley Knox, Casandra Monroe, and Michael Wigal), to appear in

*Electronic Journal of Linear Algebra*.Paperclip graphs, (with Erik D. Demaine, Martin L. Demaine, Ron Graham, Adam Hesterberg, Jason Ku, Jayson Lynch, and Tadashi Tokieda), submitted.

Properties of a

*q*-analogue of zero forcing, (with Craig Erickson, Shaun Fallat, H. Tracy Hall, Brenda Kroschel, Jephian C.-H. Lin, Bryan Shader, Nathan Warnberg, and Boting Yang), to appear in*Graphs and Combinatorics*.Spectral properties of the exponential distance matrix, (with Elizabeth Cooper, Aaron Li, Kate Lorenzen, Zoë Schopick), submitted.

Hadamard diagonalizable graphs of order at most 36 (with Jane Breen, Melissa Fuentes, Bernard Lidický, Michael Phillips, Alexander W. N. Riasanovksy, Sung-Yell Song, Ralihe R. Villagrán, Cedar Wiseman, and Xiaohong Zhang), submitted

### 2020

The mathematical life of Fan Chung,

*AMS Notices***67**(2020), 327-335.The inverse eigenvalue problem of a graph: Multiplicities and minors, (with Wayne Barrett, Shaun Fallat, Tracy Hall, Leslie Hogben, Jephian Lin, Bryan Shader, and Michael Young),

*JCTB***142**(2020), 276-306. [doi:10.1016/j.jctb.2019.10.005]A jaunt in spectral graph theory,

*50 Years of Combinatorics, Graph Theory, and Computing*, F. Chung, R. Graham, F. Hoffman, L. Hogben, R. Mullin, D. West, eds. CRC Press, 2020, 213-238.Counting tilings by taking walks in a graph, (with Jason Ekstrand and Stephen Osborne),

*A Project-Based Guide to Undergraduate Research in Mathematics*, P. E. Harris, E. Insko, A. Wootton, eds. Springer, 2020, 153-176.Enumerating parking completions using Join and Split, (with Ayomikun Adeniran, Galen Dorpalen-Barry, Pamela E. Harris, Cyrus Hettle, Qingzhong Liang, Jeremy L. Martin, Hayan Nam),

*Electronic Journal of Combinatorics***27**(2020), #P2.44, 19 pps. [doi:10.37236/9194]

### 2019

Enumerating multiplex juggling patterns, (with Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo),

*Journal of Integer Sequences***22**(2019), 19.1.7., 21 pp.Proof without words: Independent sets in grid graphs and tilings of Aztec diamonds,

*Mathematics Magazine***92**(2019), 126-127. [doi:10.1080/0025570X.2019.1542233]Graphs with at most two trees in a forest building process, (with Misa Hamanaka and Marie Hardt),

*Involve***12**(2019), 659-670. [doi:10.2140/involve.2019.12.659]On the genus of a quotient of a numerical semigroup, (with Ayomikun Adeniran, Colin Defant, Yibo Gao, Pamela E. Harris, Cyrus Hettle, Qingzhong Liang, Hayan Nam, Adam Volk),

*Semigroup Forum***98**(2019) 690-700. [doi:10.1007/s00233-018-9989-3]Graph switching, 2-ranks, and graphical Hadamard matrices, (with Aida Abiad and Willem Haemers),

*Discrete Mathematics***342**(2019), 2850-2855. [doi:10.1016/j.disc.2018.11.022].Computing Kemeny's constant for a barbell graph, (with Jane Breen, Nicklas Day, Colt DeArmond, Kate Lorenzen, Haoyang Qian, Jacob Riesen),

*Electronic Journal of Linear Algebra***35**(2019), 583-598. [doi:10.13001/1081-3810.4095]

### 2018

A forest building process on simple graphs, (with Zhanar Berikkyzy, Jay Cummings, Kristin Heysse, Paul Horn, Ruth Luo, and Brent Moran),

*Discrete Mathematics***341**(2018), 497-507. [doi:10.1016/j.disc.2017.09.017]Apollonian ring packings, (with Adrian Bolt, and Espen Hovland), in

*Connections in Discrete Mathematics*, S. Butler, J. Cooper, G. Hurlbert, eds. Cambridge University Press, 283-296.Permutations resilient to deletions, (with Noga Alon, Ron Graham, and Utkrisht Rajkumar),

*Annals of Combinatorics***22**(2018), 673-680. [doi:10.1007/s00026-018-0403-3]

### 2017

A generalization of Eulerian numbers via rook placements, (with Esther Banaian, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce),

*Involve***10**(2017), 691-705. [doi:10.2140/involve.2017.10.691]The enhanced principal rank characteristic sequence for Hermitian matrices, (with Minerva Catral, Tracy Hall, Leslie Hogben, Xavier Martinez-Rivera, Bryan Shader, and Pauline van den Driessche),

*Electronic Journal of Linear Algebra***32**(2017), 58-75. [doi:10.13001/1081-3810.3249]Zero forcing propagation time on oriented graphs, (with Adam Berliner, Chassidy Bozeman, Minerva Catral, Leslie Hogben, Brenda Kroschel, Jephian Chin-Hung Lin, Nathan Warnberg, and Michael Young),

*Discrete Applied Mathematics***224**(2017), 45-59. [doi:10.1016/j.dam.2017.02.017]Parking distributions on trees, (with Ron Graham, and Catherine Yan),

*European Journal of Combinatorics***65**(2017), 168-185. [doi:10.1016/j.ejc.2017.06.003]Juggling card sequences, (with Fan Chung, Jay Cummings, and Ron Graham),

*Journal of Combinatorics***8**(2017), 507-539. [10.4310/JOC.2017.v8.n3.a6]

### 2016

Symmetric 0-1 matrices with inverses having two distinct values and constant diagonal, (with Wayne Barrett and Tracy Hall),

*Linear Algebra and Its Applications***491**(2016), 41-55. [doi:10.1016/j.laa.2015.02.019]The enhanced principal rank characteristic sequence, (with Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, Pauline van den Driessche, and Michael Young),

*Linear Algebra and Its Applications***498**(2016), 181-200. [doi:10.1016/j.laa.2015.03.023]Inserting plus signs and adding, (with Ron Graham and Richard Stong),

*American Mathematical Monthly***123**(2016), 274-279.[doi:10.4169/amer.math.monthly.123.3.274]Algebraic aspects of the normalized Laplacian, in

*Recent Trends in Combinatorics*, Andrew Beveridge, Jerrold Griggs, Leslie Hogben, Gregg Musiker, and Prasad Tetali, eds., Springer, 2016, 295-315.The mathematics of the flip and horseshoe shuffles, (with Persi Diaconis and Ron Graham),

*American Mathematical Monthly***123**(2016), 542-556. [doi:10.4169/amer.math.monthly.123.6.542]A cospectral family of graphs for the normalized Laplacian found by toggling, (with Kristin Heysse),

*Linear Algebra and its Applications***507**(2016), 499-512. [doi:10.1016/j.laa.2016.06.033]Rainbow arithmetic progressions, (with Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard L. Kramer, Jephian Chin-Hung Lin, Ryan R. Martin, Derrick Stolee, Nathan Warnberg, and Michael Young),

*Journal of Combinatorics***7**(2016), 595-626. [doi:10.4310/JOC.2016.v7.n4.a3]Counting prime juggling patterns, (with Esther Banaian, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce),

*Graphs and Combinatorics***32**(2016), 1675-1688. [doi:10.1007/s00373-016-1711-1].

### 2015

Knight's tours on boards with odd dimensions, (with Baoyue Bi, Stephanie De Graaf, and Elizabeth Doebel),

*Involve***8**(2015), 615-627. [doi:10.2140/involve.2015.8.615]Using twins and scaling to construct cospectral graphs for the normalized Laplacian,

*Electronic Journal of Linear Algebra***28**(2015), 54-68. [doi:10.13001/1081-3810.2989]Edge flipping in the complete graph, (with Fan Chung, Jay Cummings, and Ron Graham),

*Advances in Applied Mathematics***69**(2015), 46-64. [doi:10.1016/j.aam.2015.06.002]Using variants of zero forcing to bound the inertia set of a graph, (with Jason Grout and Tracy Hall),

*Electronic Journal of Linear Algebra***30**(2015), 1-18. [doi:10.13001/1081-3810.2900]Egyptian fractions with each denominator having three distinct prime divisors, (with Paul Erdos and Ron Graham),

*INTEGERS: The Electronic Journal of Combinatorial Number Theory***15**(2015) A51, 9 pp.

### 2014

Tiling a strip with triangles, (with John Bodeen, Taekyoung Kim, Xiyuan Sun, and Shenzhi Wang),

*Electronic Journal of Combinatorics***21**(2014), P1.7, 15pp.Counting tilings by taking walks, (with Steven Osborne),

*The Journal of Combinatorial Mathematics and Combinatorial Computing***88**(2014), 83-94.The maximum nullity of a complete subdivision graph is equal to its zero forcing number, (with Wayne Barrett, Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, and Michael Young),

*The Electronic Journal of Linear Algebra***27**(2014), 444-457. [doi:10.13001/1081-3810.1629]The principal rank characteristic sequence over various fields, (with Wayne Barrett, Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, Pauline van den Driessche, and Michael Young),

*Linear Algebra and Its Applications***459**(2014), 222-236. [doi:10.1016/j.laa.2014.06.045]de Bruijn sequences with varying combs, (with Abbas Alhakim and Ron Graham),

*INTEGERS: The Electronic Journal of Combinatorial Number Theory***14A**(2014) A1, 23 pp.Unrolling residues to avoid progressions, (with Ron Graham and Linyuan Lu),

*Mathematics Magazine***87**(2014), 83-94. [doi:10.4169/math.mag.87.2.83]Numerical tic-tac-toe on the 4x4 board, (with Bryce Sandlund, Kerrick Staley, and Michael Dixon),

*COCOON 2014*, LNCS 8591, Springer-Verlag, 537-546. [doi:10.1007/978-3-319-08783-2_46]Spectral graph theory, (with Fan Chung), in

*Handbook of Linear Algebra, second edition*, Leslie Hogben ed., 14pp.A gentle introduction to the normalized Laplacian,

*IMAGE***53**(Fall 2014), 19-27.

### 2013

An interstice relationship for flowers with four petals, (with Ron Graham, Gerhard Guettler, and Colin Mallows),

*Journal of Geometry***104**(2013), 421-438. [doi:10.1007/s00022-013-0173-3]Throttling zero forcing propagation speed on graphs, (with Michael Young),

*Australasian Journal of Combinatorics***57**(2013), 65-71.Constructing points through folding and intersection, (with Erik Demaine, Ron Graham, and Tomohiro Tachi),

*International Journal of Computational Geometry & Applications***23**(2013), 49-64. [doi:10.1142/S0218195913500039]Subdivision using angle bisectors is dense in the space of triangles, (with Ron Graham),

*The American Mathematical Monthly***120**(2013), 622-630. [doi:10.4169/amer.math.monthly.120.07.622]

### 2012

Origami rings, (with Joe Buhler, Warwick de Launey, and Ron Graham),

*Journal of the Australian Mathematical Society***92**(2012), 299-311. [doi:10.1017/S1446788711001741]A note on marking lines in [k]^n, (with Ron Graham),

*Designs, Codes and Cryptography***65**(2012), 165-175. [doi:10.1007/s10623-011-9507-z]Computing inertia sets using atoms, (with Wayne Barrett, Tracy Hall, John Sinkovic, Wasin So, Colin Starr, and Amy Yielding),

*Linear Algebra and its Applications***436**(2012), 4489-4502. [doi:10.1016/j.laa.2011.08.026]

### 2011

A construction of cospectral graphs for the normalized Laplacian, (with Jason Grout)

*Electronic Journal of Combinatorics***18**(2011), #231, 20pp.Hypercube orientations with only two in-degrees, (with Joe Buhler, Ron Graham, and Eric Tressler),

*Journal of Combinatorial Theory, Series A***118**(2011), 1695-1702. [doi:10.1016/j.jcta.2011.02.003]

### 2010

Finding patterns avoiding many monochromatic constellations, (with Kevin Costello and Ron Graham),

*Experimental Mathematics***19**(2010), number 4, 399-411. [doi:10.1080/10586458.2010.10390631]Iterated triangle partitions, (with Ron Graham),

*Fete of Combinatorics and Computer Science*, G. Katona, A. Schrijver, T. Szonyi, eds., Bolyai Society Mathematical Studies 29, Springer-Verlag, Heidelberg (2010), 23-42. [doi:10.1007/978-3-642-13580-4_2]Tiling polygons with lattice triangles, (with Fan Chung, Ron Graham, and Miklos Laczkovich),

*Discrete & Computational Geometry***44**(2010), 896-903. [doi:10.1007/s00454-010-9249-0]Irreducible Apollonian configurations and packings, (with Ron Graham, Gerhard Guettler, and Colin Mallows),

*Discrete & Computational Geometry***44**(2010), 487-506. [doi:10.1007/s00454-009-9216-9]Eigenvalues of 2-edge-coverings,

*Linear and Multilinear Algebra***58**(2010), 413-423.[doi:10.1080/03081080802622694]Shuffling with ordered cards, (with Ron Graham),

*Journal of Combinatorics***1**(2010), 121-139. [doi:10.4310/JOC.2010.v1.n2.a3]Enumerating (multiplex) juggling sequences, (with Ron Graham),

*Annals of Combinatorics***13**(2010), 413-424. [doi:10.1007/s00026-009-0040-y]Small spectral gap in the combinatorial Laplacian implies Hamiltonian, (with Fan Chung),

*Annals of Combinatorics***13**(2010), 403-412. [doi:10.1007/s00026-009-0039-4]Intersecting domino tilings, (with Paul Horn and Eric Tressler),

*The Fibonacci Quarterly***48**(2010), 114-120.How to play the majority game with a liar, (with Ron Graham and Jia Mao),

*Discrete Mathematics***310**(6 February 2010), 622-629. [doi:10.1016/j.disc.2009.05.011]A note on nested sums, (with Pavel Karasik),

*Journal of Integer Sequences***13**(2010), article 10.4.4, 8 pp.A note about cospectral graphs for the adjacency and normalized Laplacian matrices,

*Linear and Multilinear Algebra***58**(2010), 387-390. [doi:10.1080/03081080902722741]

### 2009

Induced-universal graphs for graphs with bounded maximum degree,

*Graphs and Combinatorics***25**(2009), 461-468. [doi:10.1007/s00373-009-0860-x]The lost daughters of Gergonne, Forum

*Geometricorum***9**(2009), 19-26.Optimal jumping patterns, (with Ron Graham and Nan Zang),

*Journal of Combinatorics and Number Theory***1**(2009), 1-13.

### 2008

Hat guessing games, (with Mohammad Hajiaghayi, Robert Kleinberg, and Tom Leighton),

*SIAM Journal on Discrete Mathematics***22**(2008), 592-605. [doi:10.1137/060652774]; Republished in*SIAM Review***51**(2009), 399-413. [doi:10.1137/080743470]Jumping sequences, (with Ron Graham and Nan Zang),

*Journal of Integer Sequences***11**(2008), 13 pp.Estimating the number of graphs containing very long induced paths,

*Ars Combinatoria***88**(2008), 321-332.Zero forcing sets and the minimum rank of graphs, (with AIM minimum rank-special graphs work group),

*Linear Algebra and its Applications***428**(2008), 1628-1648. [doi:10.1016/j.laa.2007.10.009]

### 2007

Interlacing for weighted graphs using the normalized Laplacian,

*Electronic Journal of Linear Algebra***16**(2007), 90-98. [doi:10.13001/1081-3810.1185]Forest-like permutations, (with Mireille Bousquet-Melou),

*Annals of Combinatorics***11**(2007), 335-354. [doi:10.1007/s00026-007-0322-1]How to play the majority game with liars, (with Ron Graham and Jia Mao),

*AAIM 2007*, Lecture Notes in Computer Science 4508, Springer-Verlag, 221-230. [doi:10.1007/978-3-540-72870-2_21]

### 2006

Relating singular values and discrepancy of weighted directed graphs,

*Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm*(Miami, FL, 2006), 1112-1116. [doi:10.1145/1109557.1109680]Using discrepancy to control singular values for nonnegative matrices,

*Linear Algebra and its Applications***419**(2006), 486-493. [doi:10.1016/j.laa.2006.05.015]

### 2003

Tangent line transformations,

*The College Mathematics Journal***34**(2003), 105-106. [doi:10.2307/3595781]

### Books

*Rudiments of Ramsey Theory*, second edition, (with Ron Graham), CBMS 123, AMS press, 2015.

### Edited Volumes

*Mathematics of Paul Erdos I and II, second edition*, (with Ron Graham and Jarik Nešetřil), Springer, 2013. [doi:10.1007/978-1-4614-7258-2] [doi:10.1007/978-1-4614-7254-4]*Connections in Discrete Mathematics*, (with Joshua Cooper and Glenn Hurlbert), Cambridge, 2018.

### Miscellaneous writings and musings

Eigenvalues and structures of graphs, Ph.D dissertation, written under the direction of Fan Chung.

Bounding the number of graphs containing very long induced paths, MS Thesis, written under the direction of Wayne Barrett

Determining the underlying functions for Cauchy power and exponential forms, research done as an undergraduate student

Tangent line transformations, research done as an undergraduate student

Posed problems (Monthly #11030, #11265; Math Magazine #1668, #1730, #1761, #Q1008)

Memories about Ron Graham

The Mathematical Showman - Ron Graham (1935-2020) - Numberphile Podcast

Ron Graham has left (as Tom Odda, Jr), MAA FOCUS, Aug/Sep 2020, 8-9

In September 2006 I visited the Center for Combinatorics at Nankai University and gave a series of three lectures. The notes for these lectures are given here.

In April 2011 I gave a guest lecture about cospectral graphs at UCSD. The notes for this lecture is given here.

Some proofs without words (or not very many).

Property of the GCD; Moebius transform of the triangular numbers; Tiling Aztec diamonds with skinny strips

Various bits and pieces.

Contributions to the Gathering for Gardner over the years.

Letter to the editor of school newspaper written as an undergraduate.